We select the form (REF ) because it is the one used for studying
the substitutions of variables which keep invariant the form of the
Heun equation [1]}.
These substitutions
result from 8 homotopic transformations
(no change of the independent variable)
and 24 linear fractional transformations of the independent variable. As in the case of the hypergeometric equation, the substitutions can
also be used to transform an initial solution
into new solutions which, for the Darboux equation, can be rewritten
in terms of elliptic functions.
Thus, from an initial solution, the homotopic transformations
generate a group containing 8 expansions. New groups can be generated by firstly
applying on the initial solution a fractional transformation
of the independent variable.
